Induction, as has been said in the Introduction, is a sort of inference proceeding from particular proposition to general ones; the former being based on observation and experiment. By observation [is meant] one's attention to a certain natural phenomenon as actually occurring, to discover its causes and relations to other phenomena. By experiment is meant one's interference and effort to produce such a phenomenon in a variety of circumstances, to discover those causes and relations. The difference between observation and experiment is that between observing lightning, for instance, as it naturally occurs, and actively producing it in a certain way in the laboratory. Thus, inductive inference begins with observing a certain phenomenon or actively producing it in many cases, and then establishing a general conclusion suggested by these observations and experiments.

Aristotle did not distinguish between observation and experiment, and considered induction as any inference based on enumerating particular instances, consequently, he classified induction into perfect and imperfect, if the conclusion refers to all the particulars in question, induction is perfect, if it includes reference to some particular instances only, induction is imperfect [1].

Aristotle has considered perfect induction in a way different from his consideration of imperfect induction. Induction cannot be divided, in our view, into perfect and imperfect because induction in fact proceeds from particular to universal, whereas perfect induction does not do so, but its premises are general like its conclusion. Thus, we regard perfect induction as deduction not induction; and it is imperfect induction that is induction proper.

Aristotle's perfect induction

Perfect induction was of great logical value for Aristotle being as rigorous as syllogism. When syllogism predicates major terms of minor term by virtue of a middle term, its conclusion is certain; similarly, the conclusion of perfect induction relates a predicate to a subject by means of enumerating all instances of that subject, thus the certainty of such conclusion. Further, Aristotle considers perfect induction a basis of recognizing the ultimate premises of syllogistic reasoning.

We reach those premises not by syllogism but by perfect induction. For, in syllogism we predicate the major term to the minor term by means of the middle term, this being a predicate of the minor term and subject of the major term; and if we try to prove syllogistically that the major term is asserted of the middle term, or that the middle term is asserted of the minor term, we have to find out the middle term between them, and then we go on until we reach ultimate premises wherein we relate predicate to subject without any medium.

And as we cannot get a syllogism without a middle term, the only way for Aristotle to reach such ultimate premises of syllogism is by perfect induction. Later on, medieval logicians did not give such a great value to perfect induction, but they still regarded it as an important means of arriving at ultimate premises.

Criticism of perfect induction

Our comments on Aristotelian perfect induction are as follows:

(1) We are concerned in this book with induction proceeding from particular to universal, thus perfect induction lies outside our interest, since it is a sort of deduction the premises of which are also universal, and the principle of non contradiction is sufficient to show the truth of its conclusion.

(2) We may ask, what is the use of the conclusion of perfect induction for us? Two Aristotelian answers are expected, (i) the conclusion asserts a logical or causal relation between its two terms. When we say John, Peter and Smith are all the individuals of the human species; John, Peter and Smith eat; therefore every man eats. It may here be said that the conclusion asserts a causal relation between humanity and eating, (ii) Aristotle may not insist on regarding the conclusion as asserting a causal relation, but show the fact that men eat, by complete enumeration of all individuals.

Let us discuss these answers. Aristotle would be mistaken if he thought that perfect induction gives a causal relation between the terms of the conclusion otherwise this conclusion would give new information not included in the premises; and then the inductive reasoning loses its logical validity and cannot be explained by the law of non - contradiction alone. Further, if we take the conclusion of perfect induction as giving a fact about its terms and not a certain relation between them, such a conclusion would indeed be valid since it is contained in the premise, but then perfect induction would not be a proof in Aristotle's sense. He conceived proof as giving a logically certain relation between the terms of the conclusion, and this certainty arises from our discovery of the true cause of that relation. Such a cause may be the subject itself and the predicate may be either an essential attribute or not; if essential attribute, then the conclusion is an ultimate premise, but if not, the conclusion would be demonstrated only in a secondary sense.

Now, if the conclusion of perfect induction just states that men eat, without asserting that humanity is a cause of eating, then it is not a demonstrative proposition, and a fortiori, induction is proof no longer. And if perfect induction is unable to give demonstrative statements, then there is no way to establish ultimate premises of proof.

(3) Perfect induction gives us a judgment about, at most, actually observed instances but not instances which may exist in the future. We may observe, theoretically speaking, all the instances of man in the past and present and see that they eat, but cannot now observe men that may come in the future. Thus perfect induction cannot give us a strictly universal conclusion. And it makes no difference to make induction dealing with particulars, e.g. John, Peter .. and to arrive at a general conclusion such as every man eats, or dealing with species such as man, horse, lion to judge that all animals die. For a species or genus does not include individuals or species actually existed and observed only, but a species may have other individuals, and genus other species.

(4) Perfect induction has recently been criticized not only as a proof in the Aristotelian sense, but also as a proof in any sense. Suppose I arrived at the conclusion, all matter is subject to gravitation, after a long series of experiments in a great number of instances. Induction maybe formulated thus:

a

1' a2' a3'…an are subject to gravitation.

a

1' a2' a3'…an are all the kinds of matter that exist.

. . all matter is gravitational.

When I see a piece of stone, I judge that it is subject to this law, not because I give a new judgement, for stones are among the kinds under experiment, but because when I come across some instance not included in my experiments, I judge that the conclusion applies to the new instance as well.

This objection may be retorted on Aristotelian lines. In perfect induction, we do not intend to say that this piece or that piece of stone is subject to gravitation, but that all pieces of matter are so.

Aristotle distinguished syllogism from induction, the former predicates the major term to the minor term by means of the middle term, whereas the latter predicates the major term to the middle term by means of the minor term. Thus, the conclusion that this or that piece of stone has gravitational property is reached not by induction but by a syllogism, formulated thus: these instances have gravitational property; these instances are all matter that exists . . all matter has gravitational property

Further, it should be remarked that the statement all pieces of iron extend by heat is not merely enumerating particular statements expressing the fact this and that piece extend by heat, but it is a different statement from all those particular ones. For the statement all pieces of iron extend by heat is reached by induction in two steps. First, we collect all pieces of iron in the world, separating them from all other species of matter and conclude that these are all iron that exists. Secondly, we turn to every piece of iron and show that each extends by heat. [Only then perfect induction could be properly asserted, reader's note]

Recapitulation

The results reached so far are as follows, (a) The subject of perfect induction does not concern those who consider induction in the modern sense; (b) Perfect induction can not be regarded as a proof in the Aristotelian sense for it is unable to discover the cause; (c) Perfect induction is formally a valid inference and (d) General statements in science cannot be reached through this sort of induction.

Aristotle's imperfect induction

The Problem of induction

If you ask an ordinary man to explain how we proceed from particular statements to a general inductive conclusion, his answer may be that we face two phenomena in all experiments such as between heat and extension of iron, and since the extension of iron has a natural cause, we naturally conclude from constant relation between heat and extension that heat is the cause, and if so, we have right to make the generalization that when iron is subjected to heat it extends. But this explanation does not satisfy the logician for many reasons. (A) Induction should first establish the causal law [which is an a priori principle in rationalistic epistemology, but not in the empiricistic epistemology, which considers empirical observation to be the only source of knowledge, reader's note] among natural phenomena, otherwise extension of iron has probably no cause and may happen spontaneously, and hence another piece of iron may not extend by heat in the future. (B) If induction has got to establish causality in nature, it suggests that the extension of iron has a cause, but has no right to assert off band that the cause is heat just because heat is connected with extension. Extension of iron must have a cause but it may be something other than heat, heat might have been concomitant with the extension of iron without being its cause [since observation of two adjacent phenomena doesn't necessarily mean that one is the cause of the other, for example in the case of morning following night, nobody says night is the cause of morning, reader's note]. Induction should therefore establish that heat any other is the cause[?]. (C) If induction could establish the principle of causality among natural phenomena, and could also argue that a is the cause of b, it still has to prove that such causal relation will continue to exist in the future, and in all the yet unobserved instances, otherwise the general inductive statement is baseless [the most it could generalize is that heat causes extension in the piece(s) of iron under observation and for that piece(s) of iron only, reader's note].

Aristotelian logic has an answer on logical ground to the second question only; as to the first and the third, it is satisfied with the answers given in the Aristotelian rationalistic epistemology. Rationalism involves the causal principle (every event has a cause ), independently of sensible experience. Rationalism involves also the principle that "like causes have like effects" this being a principle deduced from causal principle, and would be the ground of the third question mentioned above. It is the second question only that the Aristotelian logic has got to face and solve, that is, how can we infer the causal relation between any two phenomena that have mere concomitance and not reduce such concomitance to mere chance? To overcome this, Aristotelian logic offers a third rationalistic principle that we now turn to state in detail.

Formal logic and the problem

When a generalisation is through induction, we either apply it to instances which are different in some properties from those we have observed, or apply it to instances that are exactly like those we have observed; the former generalisation, for Formal logic, is logically invalid, because we have no right to infer a general conclusion from premises some of which state some properties unlike the properties stated in other premises. Suppose we observed all animals and found that they move the lower part of their mouth in eating, we cannot generalise this phenomenon to sea animals, since these have different properties from the animals already observed. [2]

But inductive generalisation is logically valid, when applied to like unobserved instances which are similar to instances observed. Validity here is not based on mere enumeration of instances, for this does not prove that there is causal relation between any two phenomena. Formal logic has found a way to assert causal relation in inductive generalisations, if we add, to the observation of instances, a rational a priori principle, that is, chance cannot be permanent or repetitious, or between any two phenomena not causally related, concomitance cannot happen all the time or most of the time. Such principle may take a syllogistic form : a and b have been observed together many times, when two phenomena are observed to be severally connected, one is a cause of the other; therefore a is cause of b. This syllogism proceeds from general to particular, and not vice versa, thus not induction.

We then observe that the role played by imperfect induction, for formal logic, is producing a minor premise of a syllogism. This inductive inference involving a sort of syllogism is called by formal logicians an experience, and this is considered a source of knowledge. The difference between experience and imperfect induction is that the latter is merely an enumeration of observed instances, while the former consists of such induction plus the a priori principle already stated.

Consequently, it may be said that formal logic regards imperfect induction as a ground of science, if experience as previously defined, is added; that is if we add, to observation of several instances , the a priori principle that chance cannot happen permanently and systematically.

Misunderstanding of formal logic

Some modern thinkers mistakenly thought that formal logic rejects inductive generalisations and is interested only in perfect induction. But formal logic showed, as we have seen, that imperfect induction can give logically valid generalisation if we collected, several instances and added a rational principle, such that we reach a syllogism proving causality, and that is called experience [which is also a][xand a] source of knowledge.

Further, some commentators of formal logic have understood the distinction between imperfect induction and experience in a certain way. Perfect induction is based on passive observation while experience needs active observation. An example of the former is that when we observe a great number of all swans are black. An example of the latter is that when we heat iron and observe that iron extends and conclude that iron extends by heat. This attempt to distinguish induction from experience anticipates the modern conception of experience and makes imperfect induction similar to systematic observation. But this explanation is mistaken, for experience is meant by formal logicians no more than imperfect induction plus the construction of a syllogism, the minor premise of which is based on induction, while the major premise states a rational principle rejecting the repetition of chance happenings.

Aristotelian epistemology and induction

The formal logical view of introducing a priori principles in induction is related to rationalistic epistemology which includes that reason independently of sense experience is a source of knowledge. And this theory of knowledge is opposed to the empiricist theory which insists on sense experience as the only source of human knowledge. If we maintain that chance cannot be permanent or repetitious this must be established by induction, and thus that principle is nothing but an empirical generalisation, thus it cannot be regarded as the logical foundation of valid generalisation.

Although we are enthusiast about rationalistic epistemology, as will be shown later, we think that Aristotle's principle (chance cannot be permanent and repetitious ) is not an a priori principle, but a result of inductive process.

Formal logic and chance

Let us make clear how chance is defined by formal logicians. We may first clarify, "chance", by making clear its opposite, i.e., necessity. Necessity is either logical or empirical. Logical necessity is a relation between two statements or two collections of statements, such that if you deny one of them, then they become contradictory, e.g., logical necessity between Euclidean postulates and theorems. On the other hand, empirical necessity is a causal relation between two things such as between fire and heat, heat and boiling, poison and death; and causality has nothing to do with logical necessity, in the sense that it is not contradictory to assert that fire does not produce heat, and so on. There is a great difference between the statement' the triangle has not three side's and the statement "heat is not a cause of boiling water", The former is self contradictory while the latter is not; necessity between heat and boiling is a matter of fact not a matter of logic.

Let us now turn to chance. To say that something happens by chance is to say that it is neither logically not empirically necessary to happen. Chance is either absolute or relative. Absolute chance is the happening of something without any cause, as the boiling of water without a cause; whereas relative chance is the occurrence of an event as having a cause, but it happens that it is connected with the occurrence with another event by chance, for example, when a Kettle full of water under heat boils, but a glass of water under the zero point freezes; thus it happened by chance that the Kettle boiled at the same time when the glass freezes. Chance here is relative because both boiling and freezing have causes ( not by chance ) but their concomitance is by chance. Thus, absolute chance is the occurrence of an event without any necessity, logical or empirical - without any cause; where as relative chance is the concomitance of two events without any causal relation between them.

Now, absolute chance for Aristotle, is impossible, for this sort of chance is opposed to the causal principle. Thus, in rejecting absolute chance, Aristotelian epistemology and other sort of rationalism establish the causal principle, and consider it the basis of the answer to the first of our three questions related to the problem of induction; and goes with this the answer to the third question which is deduced from the causal principle. But, for Aristotelian rationalism, relative chance is not impossible, because it is not opposed to causality.

The concomitance between frozen water and boiled water by chance does not exclude that freezing or boiling has a cause. We have in that instance three sorts of concomitance: frozen water and boiled water, freezing, and heat to the zero point, boiling and heat in high temperature; the first being by chance, the latter two are causally related. There is a great difference between concomitance by virtue of causal relation and concomitance by relative chance; the former is uniform and repetitious, such as between the concomitance between heat and boiling, or lightning and thunder. The latter is neither uniform nor recurrent, for example, you for many times, when you go out, you meet a friend, but this does not happen uniformly.

Formal logic takes the previous view as a ground of the principle that chance does not happen permanently or uniformly, considers it a priori principle, and by chance is meant relative chance.

Need of definite formulation

Despite clear exposition previously stated, the principle that chance does not happen permanently and uniformly has to be clarified. We ought to know precisely whether the rejection of relative chance applies to all time past, present and future, or is confined to the field of experiments made by some person in a definite stretch of time.

In the former, it follows that relative chance does not recur in all time, but that is impossible since we cannot observe all natural phenomena in the past and future. And if meant by the principle that we reject uniform repetition in the field of experiments made by some person, it follows that the principle seeks to show that relative chance does not recur in a reasonable number of observations and experiments. But the Aristotelian principle has to specify the reasonable number of experiments required. Can we formulate the principle thus: relative chance does not recur in ten or hundred or thousand experiments? Suppose we specified the reasonable number by ten, then if we put some water in a low temperature and it freezes, we cannot discover the causal relation from doing the experiment only once; we have to repeat the experiment ten times, in this case we have right to discover the causal relation.

The crucial point of difference

We differ from formal logic on the principle that chance cannot happen uniformly mainly not its truth but its character.We accept the principle but refuse its being a priori and rational nature. Formal logic regards that principle as independent of all sensible experience and then is considered a ground of all inductive inferences; for if it is considered an empirical principle and derived from experience, it cannot be a principle of induction but itself an inductive generalisation. Such principle is, in our view, a result of induction, arrived at through a long chain of observations. Now, the question arises, what evidence formal logic has to maintain that such principle is a priori?

In fact, there is no evidence, and formal logic considers the principle as among primitive and primary principles and these do not need evidence or proof. Formal logic divides our knowledge into two sorts; primary and secondary; former is intuitively perceived by the mind such as the law of non-contradiction; but secondary knowledge is deduced from the primitive one, such as the internal angles of a triangle are equal to two right ones. Primitive knowledge needs no proof but secondary sort of knowledge does. But since formal logic regards experience as one of the sources of knowledge, than[?] empirical propositions are primitive.

Since formal logic regards empirical statements as primitive statements, and claims that the principle about chance is primitive, then such principle needs no demonstration, exactly as the principle of non -contradiction need not. Since we have known the definite concept of the principle which rejects relative chance for formal logic, it is now easy to reject that principle. If this Aristotelian principle asserts the impossibility of recurrence of relative chance, as the law of non -contradiction asserts the impossibility of contradiction, we can easily claim that the former principle is not found in us, because we all distinguish the law of non - contradiction from the principle of non -recurrence of relative chance. For, whereas we cannot conceive a contradiction in our world, we can conceive the uniformity of relative chance, though it does not really exist [spurious correlations in social sciences, for example, between number of fire trucks sent to rescue and the destruction caused by the fire. The more the fire trucks, it appears more the fire damage as observed in the recurring events. So is the larger number of fire truck responsible for larger destruction? There is a third variable that actually explains the cause and that is the hugeness of fire. The massive the fire, the more trucks needed every time, and the massive the fire, the more chances of destruction every time]. And if the Aristotelian principle rejects the recurrence of relative chance in our world together with admitting that it is possible to recur, then the principle is not a rational a priori principle independent of experience, because a priori principles are either necessary or impossible, if it is only possible, how can we reject it independently of sense experience? We have said enough to conclude that the principle of rejecting relative chance is not among a priori principles. In the following chapter we shall give a detailed refutation of the a priori character of the principle.

Notes:

[1]The vision of induction into perfect and imperfect does not seem to be Aristotelian, but was made by later logicians who knew perfect induction and another sort of induction which is now called intuitive induction, but imperfect induction is absent in his writing (Tr.)

[2]This is clearly stated in Avicenna's Isharat and Al-Ghazali's Criterion of Science. (These references are originally Arabic)